## Taiwanese Journal of Mathematics

### FLAT $\phi$ CURVATURE FLOW OF CONVEX SETS

David G. Caraballo

#### Abstract

For an arbitrary initial compact and convex subset $K_{0}$ of $\mathbb{R}^{n},$ and for an arbitrary norm $\phi$ on $\mathbb{R}^{n}$, we construct a flat $\phi$ curvature flow $K\left(t\right)$ such that $K\left(t\right)$ is compact and convex throughout the evolution. Previously and using similar methods, R. McCann had shown that flat $\phi$ curvature flow in the plane preserves convex, balanced sets. More recently, G. Bellettini, V. Caselles, A. Chambolle, and M. Novaga showed that flat $\phi$ curvature flow in $\mathbb{R}^{n}$ preserves compact, convex sets. We also establish a new Hölder continuity estimate for the flow. Flat $\phi$ curvature flows, introduced by F. Almgren, J. Taylor, and L. Wang, model motion by $\phi$-weighted mean curvature. Under certain regularity assumptions, they coincide with smooth $\phi$-weighted mean curvature flows given by partial differential equations as long as the smooth flows exist.

#### Article information

Source
Taiwanese J. Math., Volume 16, Number 1 (2012), 1-12.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406525

Digital Object Identifier
doi:10.11650/twjm/1500406525

Mathematical Reviews number (MathSciNet)
MR2887849

Zentralblatt MATH identifier
1241.53055

#### Citation

Caraballo, David G. FLAT $\phi$ CURVATURE FLOW OF CONVEX SETS. Taiwanese J. Math. 16 (2012), no. 1, 1--12. doi:10.11650/twjm/1500406525. https://projecteuclid.org/euclid.twjm/1500406525

#### References

• L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, 2000.
• F. J. Almgren, Jr., J. E. Taylor and L. Wang, Curvature driven flows: a variational approach, SIAM J. Control and Optimization, 31(2) (1993), 387-438.
• F. J. Almgren, Jr. and L. Wang, Mathematical existence of crystal growth with Gibbs-Thomson curvature effects, J. Geom. Anal., 10(1) (2000), 1-100.
• G. Bellettini, V. Caselles, A. Chambolle and M. Novaga, Crystalline mean curvature flow of convex sets, Arch. Rational Mech. Anal., 179 (2006), 109-152.
• G. Bellettini, V. Caselles, A. Chambolle and M. Novaga, The volume preserving crystalline mean curvature flow of convex sets in $R^{n},$ J. Math. Pures Appl., 92 (2009), 499-527.
• Y. D. Burago and V. A. Zalgaller, Geometric inequalities, Springer-Verlag, New York, 1988.
• D. G. Caraballo, A variational scheme for the evolution of polycrystals by curvature, Princeton University Ph.D. thesis, 1997.
• D. G. Caraballo, 2-dimensional flat curvature flow of crystals, Interfaces and Free Boundaries, 7(3) (2005), 241-254.
• V. Caselles and A. Chambolle, Anisotropic curvature-driven flow of convex sets, Nonlinear Analysis, 65(8) (2006), 1547-1577.
• Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci., 65(7) (1989), 207-210.
• L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, 1992.
• L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681.
• H. Federer, Geometric measure theory, Springer-Verlag, 1969.
• M. E. Gage, Curve shortening makes convex curves circular, Invent. Math., 76(2) (1984), 357-364.
• M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23(1) (1986), 69-96.
• E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser, Boston, 1984.
• M. E. Gurtin, Thermomechanics of evolving phase boundaries in the plane, Oxford University Press, 1993.
• M. E. Gurtin, Planar motion of an anisotropic interface, in Motion by Mean Curvature and Related Topics, Walter de Gruyter, 1994.
• G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.
• S. G. Krantz and H. R. Parks, The geometry of domains in space, Birkhäuser, Boston, 1999.
• S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3(2) (1995), 253-271.
• P. Mattila, Geometry of sets and measures in Euclidean spaces: fractals and rectifiability, Cambridge University Press, 1995.
• R. J. McCann, Equilibrium shapes for planar crystals in an external field, Comm. Math. Phys., 195(3) (1998), 699-723.
• S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Computational Phys., 79 (1988), 12-49.
• J. E. Taylor, J. Cahn and C. Handwerker, Geometric models of crystal growth, Acta Metall. Mater., 40 (1992), 1443-1474.
• J. E. Taylor, Mean curvature and weighted mean curvature, Acta Metall. Mater., 40 (1992), 1475-1485.
• J. E. Taylor, Motion by weighted mean curvature is affine invariant, J. Geom. Anal., 8(5) (1998), 859-864.
• R. Webster, Convexity, Oxford University Press, 1994.